# Questions tagged [monomorphisms]

This tag is for questions related to monomorphisms.

47 questions
1answer
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### Why is $V^{\vee}\otimes W^{\vee}\longrightarrow (V\otimes W)^{\vee}$ always injective?

Let $R$ be a commutative ring with $1$. For all $R$-modules $V,W$ we have a canonical $R$-linear map $V^{\vee}\otimes W^{\vee}\longrightarrow (V\otimes W)^{\vee}$ from tensor product of dual modules ...
1answer
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### Monomorphisms preserved in a pullback

The question is answered, e.g, here. Suppose $(P, p_1, p_2)$ is a pullback for $f:A\to C$ and $g:B\to C$ with $fp_1 = gp_2$. if $f$ is a monomorphism, show $p_2$ is a monomorphism. What we do is ...
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1answer
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### Are all split epimorphisms effective? [duplicate]

An epimorphism is called split if it has a section (a right inverse). An epimorphism is called effective if it has a kernel pair and is the coequalizer of its kernel pair. The ncatlab implies here ...
1answer
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### Every $\lambda-$pure morphism in a locally $\lambda-$presentable category is a regular monomorphism

Consider the page from the book by Adamek & Rosicky: Locally presentable and accessible categories. given below. I need to derive this: Here is the statement (2) to prove the universal ...
0answers
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### Proof verification: An arrow which is monic under a faithful functor is itself monic

Context: To introduce some symbols and such, what I'm seeking to prove is this: Let $F$ be a faithful functor. Suppose $F(f)$ is a monic arrow. Show $f$ is monic. This came up as part of a class ...
1answer
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### Existence of morphisms in a free completion under directed colimits,$\lambda$-accessible category

Let $\cal K$ be a $\lambda$-accessible category with directed colimits and $\cal C$ be its representative full subcategory consiting of $\lambda$-presentable objects. Let $\cal L$ be free completion ...
1answer
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### Monomorphisms, unclear basic property, Functor

Suppose that for morphisms in a category it holds that $f\circ u=v\circ f'$ and $g\circ u=v\circ g'$ and that for a functor $F$, $Fv$ is a monomorphism. Suppose that $Ff'$ and $Fg'$ are distinct. WHY ...
0answers
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### A question about linearly independent monomorphisms.

My question was too big to put it in the title: Having a set of linearly independent field monomorphisms $K→L$, is it still possible to express one of these monomorphisms as a linear combination of ...
2answers
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### How exactly is a Field monomorphism defined?

Multiple times in my question I'll use the term 'a mapping of $x$', what I mean with this is one of the element to which $x$ can be mapped to (since there could be more than one). We have Two fields, ...
3answers
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### Mono(epi)morphisms as im(sub)mersions in the category of manifolds

In the definition of the category of smooth manifolds it's usual to take the morphisms to be simply smooth functions, and this results in having monomorfisms and epimorphisms to be injective and ...
3answers
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### How do kernels prove how a function fails to be injective

I know that for $f\colon X \to Y$, where $e_Y$ is the identity of $Y$: $$\ker(f) = \left\{x \in X \, \middle| \, f(x) = e_Y \right\}$$ I've learnt that kernels imply how much a homomorphism fails ...
2answers
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### Is $\mathbb Z_7$ an extension of $\mathbb Z_2$?

Is $\mathbb Z_7$ a extension of $\mathbb Z_2?$ I know it is not, because we can't establish a monomorphism between $\mathbb Z_7$ and $\mathbb Z_2$,i.e. it doesn't exist an injective homomorphism ...
2answers
524 views

### Any category with zero object, every kernel is monomorphism.

This is the problem 5.53 of Rotman, Homological Algebra. In any category having a zero object calling it $0$, prove that every kernel is monomorphism. Let $f:A\to B$ and $\ker(f):K\to A$ be the ...
1answer
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### Proof that f is a monomorphism [closed]

let $f:A \rightarrow B$ and $g:B \rightarrow A$ given that $\;\;g \circ f = 1_A$ how can I prove that f is a monomorphism? (I'm used to prove monomorphism in a different way, so I'm kind of lost)
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1answer
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### Categorical way of making monos commute

Is there a categorical way (in terms of diagrams, limits, lifting properties etc) to formulate the requirement that for every pair of monos $f,g:C \to D$ there should be an endomorphism $h:D \to D$, s....
1answer
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### Are Monomorphisms injective?

In the categories of topological spaces, rings, groups and sets I know that a morphism is a monomorphism iff it's injective. Things are different for schemes. In fact I know that a scheme injective ...
0answers
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### Monomorphisms and injectivity predicates

This is a curiosity question which struck me at a less-than-optimal moment; I apologize for not having thought much about it. Motivation. It is well-known that monomorphisms in a concrete category ...
0answers
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### Monomorphism preservation by pullback

I'm working through Category theory for the sciences (by David Spivak) and I'm a bit stuck in section 2 - my problem is with Proposition 2.7.5.5 and Exercise 2.7.5.6, I hope someone can help :) The ...
1answer
147 views

### Does this property characterize monomorphisms?

Is requiring that $f:\mathrm{Hom}(A,B)$ is mono the same as requiring that the pullback $A\times_BA$ of $f$ along itself is isomorphic to $A$? In Sheaves in Geometry and Logic, I read the following ...
1answer
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### Not understanding a line in a proof concerning Monomorphism and injectivity

In the proof that "in the category Div of divisible (abelian) groups and group homomorphisms between them there are monomorphisms that are not injective" given in Wikipedia http://en.wikipedia.org/...